Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities
Combinatorics
2021-01-01 v2 Quantum Algebra
Rings and Algebras
Representation Theory
Abstract
We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices.
Keywords
Cite
@article{arxiv.0809.3516,
title = {Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities},
author = {Sergio Caracciolo and Andrea Sportiello and Alan D. Sokal},
journal= {arXiv preprint arXiv:0809.3516},
year = {2021}
}
Comments
LaTeX2e, 43 pages. Version 2 corrects an error in the statements of Propositions 1.4 and 1.5 (see new Remarks in Section 4) and includes a Note Added at the end of Section 1 comparing our work with that of Chervov et al (arXiv:0901.0235)